Voronoi Diagrams Computational Geometry, WS 2006/07 Lecture 10 Prof. Dr. Thomas Ottmann Algorithmen...

Voronoi Diagrams

Computational Geometry, WS 2006/07Lecture 10

Prof. Dr. Thomas Ottmann

Algorithmen & Datenstrukturen, Institut für InformatikFakultät für Angewandte WissenschaftenAlbert-Ludwigs-Universität Freiburg

Computational Geometry, WS 2006/07Prof. Dr. Thomas Ottmann 2

Overview

• Motivation• Voronoi definitions• Characteristics• Size and storage• Construction

– Divide and Conquer

– Fortune’s Algorithm

• Applications


The Voronoi Diagram

Viewpoint 1: Locate the nearest dentist.Viewpoint 2: Find the ‘service area’ of potential customers for each dentist.


The Voronoi Diagram


Voronoi Regions

Eucledian distance :

dist(p,q) := ²² yyxx qpqp

Let P :={ p1, p2, ...,pn } be a set of n distinct points in a plane.We define the voronoi diagram of P as the subdivision of the planeinto n cells, with the property that a point q lies in the cell correspon-ding to a site pi iff dist(q, pi ) < dist(q, pj ) for each pj P with j i.

We denote the Voronoi diagram of P by Vor(P).The cell that corresponds to a site pi is denotd by V(pi ), called thevoronoi cell of pi.


Example

V(pi) = 1 j n, j i h(pi, pj)

q4

q1

q3

q2

p


Computing the Voronoi Diagram

Input: A set of points (sites) Output: A partitioning of the plane into regions of equal nearest neighbors


Voronoi Diagram Animations

Java applet animation of the Voronoi Diagram by: Christian Icking, Rolf Klein, Peter Köllner, Lihong Ma (FernUniversität Hagen)


Characteristics of the Voronoi Diagram

(1) Voronoi regions (cells) are bounded by line segments.

Special case :

Collinear points

Theorem : Let P be a set of n points (sites) in the plane. If all the sites are collinear, then Vor(P) consist of n-1 parallel lines and n cells. Otherwise, Vor(P) is a connected graph and its edges are either line segments or half-lines.

e

pi pj

pk

h(pi,pj)

h(pj,pk)If pi, pj are not collinear with pk, thenh(pi, pj ) and h(pj, pk ) can not be parallel!


Vor(P) is Connected

If Vor(P) is not connected then there would be a Voronoicell V(Pi ) splitting the plane into two halfes. Because Voronoi cells are convex, V(Pi ) would consist of a strip bounded by two parallel full lines, but we know that edges of Voronoi diagram cannot be full lines, hence a contradiction.

Claim: Vor(P) is connected

Proof by contradiction:


Other Characteristics; (2),(3)

Assumption: No 4 points are on the circle.

(2) Each vertex (corner) of VD(P) has degree 3

(3) The circle through the three points defining a Vertex of the Voronoi diagram does not contain any further point


Other Characteristics; (4),(5)

(4) Each nearest neighbor of one point defines an edge of the Voronoi region of the point.

(5) The Voronoi region of a point is unbounded iff the point lies exactly on the convex hull of the point set.


Size and Storage

Size of the Voronoi Diagram:

V(p) can have O(n) vertices!


Size of the Voronoi Diagram

Theorem: The number of vertices in the Voronoi diagram of a set of n points in the plane is at most 2n-5 and the number of edges is at most 3n-6.

Proof: 1. Connect all Half-lines with fictitious point 2. Apply Euler`s formula:v – e + f = 2

For VD(P) + : v = number of vertices of VD(P) + 1e = number of edges of VD(P) f = number of sites of VD(P) = n

Each edge in VD(P) + has exactly two vertices and each vertex of VD(P) + has at least a degree of 3:

sum of the degrees of all vertices of Vor(P) +

= 2 · ( # edges of VD(P) )

3 · ( # vertices of VD(P) + 1 )


Proof (Continued)

Number of vertices of VD(P) = vp

Number of edges of VD(P) = ep

We can apply: (vp + 1) – ep + n = 2

2 ep 3 (vp + 1)

2 ep 3 ( 2 + ep - n)

= 6 + 3ep – 3n

3n – 6 ep


Example


Storage Issues

1

1

3

4

5

6

2

2 3

4

5

•

•

•

•

Three Records (DCEL):

vertex {CoordinatesIncident edge

}; face {

OuterComponentInnerComponents

}; halfedge {

OriginTwinIncidentFaceNextPrev

};

e.g. :

Vertex 1 = {(1,2) | 12}Face 1 = {15 | [] }Edge 54 = { 4 | 45 | 1 | 43 | 15 }


Computing the Voronoi Diagram

Input: A set of points (sites) Output: A partitioning of the plane into regions of equal

nearest neighbors.


Divide and Conquer: Divide Step

Input: A set of points (sites) Output: A partitioning of the plane into regions of equal

nearest neighbors.

Divide: Divide the point set into two halves


Divide and Conquer: Conquer Step

Conquer: Recursively compute the Voronoi diagrams for the smaller point sets.

Abort condition: Voronoi diagram of a single point is the entire plane.


Divide and Conquer: Merge

Merge the diagrams by a (monotone) sequence of edges)


The Result

The finished Voronoi Diagram

Running time: With n given points is O(n log n)


Geometrical Divide and Conquer

Problem: Determine all intersecting pairs of segments

A

BC

DE

A

BC

DE

S1 S2

S

EC

D

A

B


DAC: Construction of the Voronoi Diagram

Divide:Divide P by a vertical dividing line T into 2 equal size subsets say P1 and P2. If |P| = 1 completed.

Conquer:Compute VD(P1 ) and VD(P2 ) recursively.

Merge: Compute the edge sequence K separating P1 and P2 Cut VD(P1) and VD(P2 ) by means of K starting from VD(P1 ) and VD(P2 ) and K

P1

P2

T

Theorem: If K can be computed in time O(n), then the running time of the D&C-algorithm is T(n) = O(n log n)

Proof : T(n) = 2 T(n/2) + O(n), T(1) = O(1)


Computation of K

P1 P2

First edge in K

Last edge in K

4 tangential points P1 P2 Observation: K is y - monotonous

Determines intersection s1 of m with Vor(p1) below l Determines intersection s2 of m with Vor(p2) below l

Incremental (sweep line) construction(p1 in P1 and p2 in P2 perpendicular with m, Sweep l)

Extend K by line segment l si

Set l = si

Compute new K defining pair p1, p2

Theorem: Running time O(n) Proof: Vor(pi) are convex, therefore each one‘s forward - edge are only visited once.


Example


Fortune’s Algorithm

Beach - line

Sweep - line

Observations: Intersection of the parabolas define edgesNew "telephones" ( ) define new parabolas Parabola intersection disappear, if C(P, q) has 3 points


Utilizing the Voronoi Diagram

Search for nearest neighbour• Input: A fixed (static) set P of n points in the plane, a query point p• Output: A nearest neighbour of p in P

Solution• Construct the Voronoi diagram for P in time O(n log n)• Solve the point location problem.


Enclosing Triangle


Use (Dynamic Object Set)

Search for next neighbour :Idea : Hierarchical subdivision of VD(P) Step 1 : Triangulation of final Voronoi regions Step 2 : Summary of triangles and structure of a search tree

Rule of Kirkpatrick : Remove in each case points with degree < 12,its neighbor is already far.

Theorem: Using the rule of Kirkpatrick a search tree of logarithmic depth develops.

A a

b c a b c

A


Utilizing the Voronoi Diagram

Search for nearest neighbour• Input: A fixed (static) set P of n points in the plane, a query point p• Output: A nearest neighbour of p in P

Solution• Construct the Voronoi diagram for P in time O(n log n)• Solve the point location problem.

Theorem: For a given set of n points we can construct a structure in time O(n log n) such that a nearest neighbour query can be answered in time O(log n).


Use (Static Point Set)

Closest pair of points: Go through edge list for VD(P) and determine minimum

All next neighbors :Go through edge list for VD(P) for all points and get next neighbors in each case

Minimum Spanning tree (after Kruskal)

Theorem: The MST can be computed in time O(n log n)


MST (after Kruskal): Informal Algorithm

Minimum Spanning Tree (after Kruskal):

Construct for a graph G = (V, E) a minimum spanning tree in time O(|E| log |E|).

1. Each point p from P defines 1-node tree; start with the forest of these 1-node trees.

2. If there are more than one tree T in the current forest, 2.1) find p, p´ with p in T and p´ not in T with d(p, p´)

minimal. 2.2) connect T containing p and T´ containing p´

(union of T and T´)


Kruskal’s Algorithm

Minimum Spanning Tree (after Kruskal):

Manipulates a forest of trees.

Find(v) returns the tree to which node v belongs.

Union(v, w) merges two trees with names v and w to a new one with the name w.

Make-set(v) returns the tree which has v as its only node.


Formal AlgorithmMinimum Spanning Tree (after Kruskal): G = (V, E).• Initialise E´ as empty set of edges.• Sort the edges in E according to increasing lengths.

For all v V do Make-set(v);

For all (v, w) E in increasing length order doif Find(v) Find(w) then {Select edge (v, w):}

{ Union (Find(v), Find(w)); E´ = E + {(v, w)} }

Naïve Implementation: E is the set of all n(n-1)/2 edges (v, w) with points v and w in P.


Summary

Minimum Spanning Tree (after Kruskal)

Crucial Observation: It is sufficient to consider the O(n) distances between (nearest) neighbours in order to find a pair of points with minimal distance!

It suffices to apply Kruskal’s algorithm to the dual graph of the Voronoi diagram with only O(n) edges!

Theorem: If the Voronoi diagram for a set of n points is known, the minimum spaning tree can be constructed in time O(n log n).

Voronoi Diagrams Computational Geometry, WS 2006/07 Lecture 10 Prof. Dr. Thomas Ottmann Algorithmen...

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